# (1, k)-Swap Local Search for Maximum Clique Problem

## TL;DR

This paper introduces a modified local search algorithm for the Maximum Clique Problem that efficiently improves solutions by considering replacements of multiple vertices simultaneously, demonstrating effectiveness on standard benchmark datasets.

## Contribution

The paper presents a novel (1, k)-swap local search algorithm capable of improving maximal solutions in polynomial time for the MCP.

## Key findings

- Effective on standard benchmark datasets
- Can determine in polynomial time if a solution can be improved by replacing multiple vertices
- Shows promising results on various graph instances

## Abstract

Given simple undirected graph G = (V, E), the Maximum Clique Problem(MCP) is that of finding a maximum-cardinality subset Q of V such that any two vertices in Q are adjacent. We present a modified local search algorithm for this problem. Our algorithm build some maximal solution and can determine in polynomial time if a maximal solution can be improved by replacing a single vertex with k, k > 1, others. We test our algorithms on DIMACS[5], Sloane[15], BHOSLIB[1], Iovanella[8] and our random instances.

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Source: https://tomesphere.com/paper/1704.00908